EXTENSIONS OF REGULAR ORTHOGROUPS BY INVERSE SEMIGROUPS

1995 ◽  
Vol 05 (03) ◽  
pp. 317-342 ◽  
Author(s):  
BERND BILLHARDT
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Normal Extension ◽  
Group Congruence ◽  
Completely Regular ◽  
Orthodox Semigroups

Let V be a variety of regular orthogroups, i.e. completely regular orthodox semigroups whose band of idempotents is regular. Let S be an orthodox semigroup which is a (normal) extension of an orthogroup K from V by an inverse semigroup G, that is, there is a congruence ρ on S such that the semigroup ker ρ of all idempotent related elements of S is isomorphic to K and S/ρ≅G. It is shown that S can be embedded into an orthodox subsemigroup T of a double semidirect product A**G where A belongs to V. Moreover T itself can be chosen to be an extension of a member from V by G. If in addition ρ is a group congruence we obtain a recent result due to M.B. Szendrei [16] which says that each orthodox semigroup which is an extension of a regular orthogroup K by a group G can be embedded into a semidirect product of an orthogroup K′ by G where K′ belongs to the variety of orthogroups generated by K.

scholarly journals Semigroups whose idempotents form a subsemigroup

1990 ◽  
Vol 41 (2) ◽  
pp. 161-184 ◽  
Author(s):  
Jean-Camille Birget ◽  
Stuart Margolis ◽  
John Rhodes
Keyword(s):  
Inverse Semigroup ◽  
Finite Semigroup ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Type Ii ◽  
Finite Semigroups ◽  
Case Type ◽  
Orthodox Semigroups

We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.For these three classes of semigroups, type-II is equal to type-II construct.

scholarly journals On quasi-F-orthodox semigroups

1995 ◽  
Vol 38 (3) ◽  
pp. 361-385 ◽  
Author(s):  
Bernd Billhardt ◽  
Mária B. Szendrei
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Band Variety ◽  
Orthodox Semigroups

An orthodox semigroup S is termed quasi-F-orthodox if the greatest inverse semigroup homomorphic image of S1 is F-inverse. In this paper we show that each quasi-F-orthodox semigroup is embeddable into a semidirect product of a band by a group. Furthermore, we present a subclass in the class of quasi-F-orthodox semigroups whose members S are embeddable into a semidirect product of a band B by a group in such a way that B belongs to the band variety generated by the band of idempotents in S. In particular, this subclass contains the F-orthodox semigroups and the idempotent pure homomorphic images of the bifree orthodox semigroups.

JOINS OF INVERSE SEMIGROUP VARIETIES AND BAND VARIETIES

1991 ◽  
Vol 01 (03) ◽  
pp. 371-385 ◽  
Author(s):  
PETER R. JONES ◽  
PETER G. TROTTER
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Lattice Of Varieties ◽  
Orthodox Semigroups ◽  
Semigroup Varieties

The joins in the title are considered within two contexts: (I) the lattice of varieties of regular unary semigroups, and (II) the lattice of e-varieties (or bivarieties) of orthodox semigroups. It is shown that in each case the set of all such joins forms a proper sublattice of the respective join of the variety I of all inverse semigroups and the variety B of all bands; each member V of this sublattice is determined by V ∩ I and V ∩ B. All subvarieties of the join of I with the variety RB of regular bands are so determined. However, there exist uncountably many subvarieties (or sub-bivarieties) of the join I ∨ B, all of which contain I and all of whose bands are regular.

scholarly journals Identities for existence varieties of regular semigroups

1989 ◽  
Vol 40 (1) ◽  
pp. 59-77 ◽  
Author(s):  
T.E. Hall
Keyword(s):  
Type Theorem ◽  
Inverse Semigroups ◽  
Direct Products ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Existence Variety ◽  
Completely Regular Semigroups ◽  
Locally Inverse Semigroups ◽  
Orthodox Semigroups ◽  
Regular Rings

A natural concept of variety for regular semigroups is introduced: an existence variety (or e-variety) of regular semigroups is a class of regular semigroups closed under the operations H, Se, P of taking all homomorphic images, regular subsernigroups and direct products respectively. Examples include the class of orthodox semigroups, the class of (regular) locally inverse semigroups and the class of regular E-solid semigroups. The lattice of e-varieties of regular semigroups includes the lattices of varieties of inverse semigroups and of completely regular semigroups. A Birkhoff-type theorem is proved, showing that each e-variety is determined by a set of identities: such identities are then given for many e-varieties. The concept is meaningful in universal algebra, and as for regular semigroups could give interesting results for e-varieties of regular rings.

scholarly journals A note on congruences on orthodox semigroups

1985 ◽  
Vol 26 (1) ◽  
pp. 25-30
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Orthodox Semigroup ◽  
Explicit Construction ◽  
Lattice Of Congruences ◽  
Orthodox Semigroups

C. Eberhart and W. Williams [3] showed that the least inverse semigroup congruence , on an orthodox semigroup S, plays a very important role in determining the structure of the lattice of congruences on S. In this note we show that their results can be applied to give an explicit construction for the idempotent separating congruences on S in terms of idempotent separating congruences on S/.

scholarly journals The $K$-Theory of Some Reduced Inverse Semigroup $C^*$-Algebras

2015 ◽  
Vol 117 (2) ◽  
pp. 186 ◽  
Author(s):  
Magnus Dahler Norling
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Inverse Semigroups ◽  
Crossed Product ◽  
Crossed Products ◽  
One Dimensional ◽  
C Algebras ◽  
Morita Equivalent ◽  
K Theory

We use a recent result by Cuntz, Echterhoff and Li about the $K$-theory of certain reduced $C^*$-crossed products to describe the $K$-theory of $C^*_r(S)$ when $S$ is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that $C^*_r(S)$ is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

Boolean Congruence Lattices of Orthodox Semigroups

1991 ◽  
Vol 43 (2) ◽  
pp. 225-241 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Commutative Semigroups ◽  
Completely Regular ◽  
Clifford Semigroups ◽  
Congruence Lattices ◽  
Completely Regular Semigroups ◽  
Orthodox Semigroups

The problem of characterizing the semigroups with Boolean congruence lattices has been solved for several classes of semigroups. Hamilton [9] and the author of this paper [1] studied the question for semilattices. Hamilton and Nordahl [10] considered commutative semigroups, Fountain and Lockley [7,8] solved the problem for Clifford semigroups and idempotent semigroups, in [1] the author generalized their results to completely regular semigroups. Finally, Zhitomirskiy [19] studied the question for inverse semigroups.

On left quasinormal orthodox semigroups

1983 ◽  
Vol 95 (1-2) ◽  
pp. 59-71 ◽  
Author(s):  
Gracinda M. S. Gomes
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Orthodox Semigroup ◽  
Local Isomorphism ◽  
Orthodox Semigroups

SynopsisThe existence of a smallest inverse congruence on an orthodox semigroup is known. It is also known that a regular semigroup S is locally inverse and orthodox if and only if there exists a local isomorphism from S onto an inverse semigroup T.In this paper, we show the existence of a smallest R-unipotent congruence ρ on an orthodox semigroup S and give its expression in the case where S is also left quasinormal. Finally, we prove that a regular semigroup S is left quasinormal and orthodox if and only if there exists a local isomorphism from S onto an R-unipotent semigroup T.

ON E-UNITARY COVERS OF ORTHODOX SEMIGROUPS

1993 ◽  
Vol 03 (03) ◽  
pp. 317-333 ◽  
Author(s):  
MÁRIA B. SZENDREI
Keyword(s):  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Regular Semigroups ◽  
Band Variety ◽  
Orthodox Semigroups

In this paper we prove that each orthodox semigroup S has an E-unitary cover embeddable into a semidirect product of a band B by a group where B belongs to the band variety generated by the band of idempotents in S. This result is related to an embeddability question on E-unitary regular semigroups raised previously.

scholarly journals Proper Dubreil-Jacotin inverse semigroups

1975 ◽  
Vol 16 (1) ◽  
pp. 40-51 ◽  
Author(s):  
R. McFadden
Keyword(s):  
Inverse Semigroup ◽  
Algebraic Structure ◽  
Inverse Semigroups ◽  
Partial Ordering ◽  
Group Congruence ◽  
Complete Class ◽  
Ordered Semigroups ◽  
Partially Ordered ◽  
Minimum Group ◽  
Integrally Closed

This paper is concerned mainly with the structure of inverse semigroups which have a partial ordering defined on them in addition to their natural partial ordering. However, we include some results on partially ordered semigroups which are of interest in themselves. Some recent information [1, 2, 6, 7,11] has been obtained about the algebraic structure of partially ordered semigroups, and we add here to the list by showing in Section 1 that every regular integrally closed semigroup is an inverse semigroup. In fact it is a proper inverse semigroup [10], that is, one in which the idempotents form a complete class modulo the minimum group congruence, and the structure of these semigroups is explicitly known [5].

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